A new and improved quantitative recovery analysis for iterative hard thresholding algorithms in compressed sensing

We present a new recovery analysis for a standard compressed sensing algorithm, Iterative Hard Thresholding (IHT) (Blumensath and Davies, 2008), which considers the fixed points of the algorithm. In the context of arbitrary measurement matrices, we derive a sufficient condition for convergence of IHT to a fixed point and a necessary condition for the existence of fixed points. These conditions allow us to perform a sparse signal recovery analysis in the deterministic noiseless case by implying that the original sparse signal is the unique fixed point and limit point of IHT, and in the case of Gaussian measurement matrices and noise by generating a bound on the approximation error of the IHT limit as a multiple of the noise level. By generalizing the notion of fixed points, we extend our analysis to the variable stepsize Normalised IHT (N-IHT) (Blumensath and Davies, 2010). For both stepsize schemes, we obtain asymptotic phase transitions in a proportional-dimensional framework, quantifying the sparsity/undersampling trade-off for which recovery is guaranteed. Exploiting the reasonable average-case assumption that the underlying signal and measurement matrix are independent, comparison with previous results within this framework shows a substantial quantitative improvement

A Nystr\"om method with missing distances

We study the problem of determining the configuration of $n$ points, referred to as mobile nodes, by utilizing pairwise distances to $m$ fixed points known as anchor nodes. In the standard setting, we have information about the distances between anchors (anchor-anchor) and between anchors and mobile nodes (anchor-mobile), but the distances between mobile nodes (mobile-mobile) are not known. For this setup, the Nystr\"om method is a viable technique for estimating the positions of the mobile nodes. This study focuses on the setting where the anchor-mobile block of the distance matrix contains only partial distance information. First, we establish a relationship between the columns of the anchor-mobile block in the distance matrix and the columns of the corresponding block in the Gram matrix via a graph Laplacian. Exploiting this connection, we introduce a novel sampling model that frames the position estimation problem as low-rank recovery of an inner product matrix, given a subset of its expansion coefficients in a special non-orthogonal basis. This basis and its dual basis--the central elements of our model--are explicitly derived. Our analysis is grounded in a specific centering of the points that is unique to the Nystr\"om method. With this in mind, we extend previous work in Euclidean distance geometry by providing a general dual basis approach for points centered anywhere.Comment: 11 page

Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in X-ray CT

Iterative image reconstruction (IIR) with sparsity-exploiting methods, such as total variation (TV) minimization, investigated in compressive sensing (CS) claim potentially large reductions in sampling requirements. Quantifying this claim for computed tomography (CT) is non-trivial, because both full sampling in the discrete-to-discrete imaging model and the reduction in sampling admitted by sparsity-exploiting methods are ill-defined. The present article proposes definitions of full sampling by introducing four sufficient-sampling conditions (SSCs). The SSCs are based on the condition number of the system matrix of a linear imaging model and address invertibility and stability. In the example application of breast CT, the SSCs are used as reference points of full sampling for quantifying the undersampling admitted by reconstruction through TV-minimization. In numerical simulations, factors affecting admissible undersampling are studied. Differences between few-view and few-detector bin reconstruction as well as a relation between object sparsity and admitted undersampling are quantified.Comment: Revised version that was submitted to IEEE Transactions on Medical Imaging on 8/16/201

Gender, ethnicity and teaching evaluations : Evidence from mixed teaching teams

This paper studies the effect of teacher gender and ethnicity on student evaluations of teaching quality at university. We analyze a unique data-set featuring mixed teaching teams and a diverse, multicultural, multi-ethnic group of students and teachers. Co-teaching allows us to study the impact of teacher gender and ethnicity on students’ evaluations of teaching exploiting within course variation in an empirical model with course-year fixed effects. We document a negative effect of being a female teacher on student evaluations of teaching, which amounts to roughly one fourth of the sample standard deviation of teaching scores. Overall women are 11 percentage points less likely to attain the teaching evaluation cut-off for promotion to associate professor. The effect is robust to a host of co-variates such as course leadership, teacher experience and research quality. There is no evidence of a corresponding ethnicity effect. Our results point to an important gender bias and indicate that the use of teaching evaluations in hiring and promotion decisions may put female lectures at a disadvantage